Mathematics for the interested outsider

UPDATE: I originally phrased my closing sentence poorly, and have chosen to revise it. The original should remain, struck through

I’d been hoping to write about today’s entry in the department’s graduate student seminar today, but there wasn’t a speaker lined up. So I guess I’ll have to get my rant on again.

I spoke recently with a young graduate student in a math department who’d come from undergraduate study in physics. I like to draw connections between my interests and whatever else I come across, so I mentioned John Baez’ work on “higher gauge theory”, which takes the very geometric approach to particle physics known as gauge theory and “categorifies” it. It’s a very interesting program, and I hope to talk more about it here.

The student, however, waved his hands and dismissed all discussion. I pointed out that the paper Baez wrote with Urs Schreiber — who run The n-Category Café along with David Corfield — gives a beautiful explanation of a thereto ad hoc “fake flatness” condition in other approaches to extending gauge theory. I know from conversations with other mathematicians this student had spoken to that he’s interested in just this sort of extension, so why the complete disinterest? “String theory has already done all this.”

String theory, as many popular articles and books have explained, moves from the basic premise that instead of being little points the fundamental particles are all tiny vibrating “strings” of energy. From this hypothesis flows a vast theoretical model that many people think will ultimately be the “Theory of Everything” uniting all of fundamental physics the way Newton’s theory of gravity united terrestrial and celestial mechanics so long ago.

The operative word here, though, is think. String theory provides a sweeping vision of physics and a rich loam of mathematical conjectures, but the vision remains on the horizon and the conjectures stand unproven. “No matter,” say the string theorists, “It’s obvious that the math works this way because this is how it should work.” Except mathematics doesn’t work with “should”.

Here is the metaphor I return to again and again in conversation: a mathematician and a physicist walk along a sidewalk and come upon a crack in the pavement. The physicist notes the crack is only a millimeter wide and steps over it without a second thought. The mathematician (possibly afflicted with ADHD) decides to stop and peer into the crack, finding it to be many kilometers deep. The physicist rushes ahead and makes many grand predictions which turn out to be true with startling accuracy while the mathematician lags behind, carefully plastering up the crack.

This is all well and good, and the accurate predictions of physical theories like quantum electrodynamics (“QED”) have more than justified pushing ahead while the foundations are shored up from behind. String theory, though, suffers from the fact that it makes no new predictions, and has even ventured into an intellectual territory known as “the landscape” which leaves so many choices that no real predictions are possible. It’s like arguing with an astrologer. If one choice is refuted there are literally millions of infinities of alternate choices for them to fall back on. All that’s left is a fertile ground of mathematical conjecture.

So here’s where the problem really begins: string theory has grown like kudzu, and is choking the life out of fundamental physics. The more popular it gets the harder it is to get funding for alternative theories like loop quantum gravity (so much for the government’s commitment to presenting all sides of a scientific debate). The harder it is to get funding, the less likely it is that new physicists will go into those areas. The fewer new physicists entering alternate fields the more popular string theory becomes. And around and around and around in a vicious circle. The feedback loop is even used by some string theorists as a kind of evidence! “See what a large fraction of the papers on the high-energy physics theory arXiv deal with string theory? Obviously it’s correct if so many people believe it!”

And here’s where it really gets sinister: since all that’s left in string theory is a fertile ground of mathematical conjectures, some string theory zealots are turning their eyes on mathematics to invade. “Peer review is so slow and holds up progress. All papers should just be put on the math arXiv and forget the journals.” Who cares about being able to check someone’s work? Who cares about the well-known effect in journalism that few people who read today’s big story ever hear about tomorrow’s retraction?

One of the scariest discussions I ever had with a young, gung-ho string theorist was about the Atiyah-Singer Index Theorem. This is possibly one of the most beautiful results of the 20th century, and one of the more complicated to prove. About 25 years after it was first proved a “string theory proof” emerged, which was much shorter and established most cases physicists care about. I’ve seen such a “proof” of ASIT and while it did fit onto three pages it was riddled with holes from a mathematician’s perspective. In the discussion I was using this contrast as an example of how mathematical results suggested by string theory could be much more difficult to rigorously pin down. His response chilled me: “maybe it’s time mathematics started accepting string theory proofs as valid.”

String theory is a fascinating model and a wellspring of mathematical conjecture, but as it stands it is not physics. Those who rabidly hold the party line that string theory is a — is the — model of fundamental physics use that as an axiom and will not hear dissent. I’ve seen enough electronic blood shed at Peter Woit’s Not Even Wrong and his main vilifier’s site (to which I will not link) to know that there are many string theorists who have completely drained their Kool-Aid and become fanatics in the service of this ideology.

And so string theory has gonerisks going from a possible model of fundamental particles to a close-minded fundamentalist physics.

I just got another email from the school I mentioned last Thursday. This one still didn’t say the magic words, “short list”, but it basically seems to say the same thing.

I was on the market last year as well as I finished up my degree. Partly I’m still here because topology was relatively tight last year, and partly because I didn’t send out nearly enough applications. In particular, I didn’t apply for tenure-tracks though I really should have. Yale’s department is wonderful for teaching mathematics, not so much for teaching all this annoying business of actually being a mathematician (in the professional sense).

Anyhow, to hear even these two little echoes from a school is more than I heard back last year. The whole hiring culture seems geared towards not telling anyone anything until you have to, and sometimes not even then. Still, while this place wouldn’t be my first choice it does have some things going for it. Here’s hoping.

Evidently the influx from those links in the other day pushed me to number 79 in the top blogs on WordPress for yesterday. Of course, today (which in UTC is already over) saw a fall-off, familiar to anyone who watched a tech stock’s IPO back in the ’90s. Of course, Abstract Nonsense has been around longer and sits at a stable 69.

Overshoot and fall-off.. feedback chase curve or Gibbs phenomenon.. There’s much to be posted about yet.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.